“”YC'rC Ion III an") Ian "H‘ll ABSTRACT ON THE DIFFRACTION OF PLANE ELECTROMAGNETIC WAVES BY AN INFINITE SLIT by Robert J. Spahn The problem of the diffraction of plane polarized electro- magnetic waves incident normally on an infinite slit of finite width is solved by the use of the Lebedev integral transform and the Wiener-Hopf techniqne. In particular, an expression for the ratio of the transmitted energy per unit area to the incident energy per unit area (transmission coefficient) is obtained for a <<9\, where a is one- half of the slit width and ‘A is the wavelength. ON THE DIFFRACTION OF PLANE ELECTROMAGNETIC WAVES BY AN INFINITE SLIT By Robert Joseph Spahn A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1963 G amas 7/£/’-b‘~f ACKNOWLEDGMENT The author wishes to thank Dr. Alfred Leitner for suggesting the problem and for helpful discussions throughout the course of its solution. His constant encouragement, criticisms, and assistance are greatly appreciated. ii TABLE OF CONTENTS Chapter I. INTRODUCTION II. STATEMENT OF THE PROBLEM A. Boundary Conditions B. Discussion of the Boundary Conditions C. Integral Representation of the Scattered Field D. Lebedev-Kontorovich Integral Transform Theorem III. APPLICATION OF THE BOUNDARY CONDITIONS AND A DISCUSSION OF THE RESULTING INTEGRAL EQUATIONS A. Properties ofAUA) B. Growth «Am C. Validity of the Applications of Lebedev's Theorem IV. SOLUTION OF THE PROBLEM A. The Dual Integral Equations and the Wiener-Hapf Technique iii Page Ul-F'W 11 16 19 21 21 TABLE OF CONTENTS, CONTINUED Chapter B. The Scattered Field Expressed as a Series Expansion C. The Transmission Coefficient V. DISCUSSION AND CONCLUSIONS REFERENCES iv Page 30 51 1m 45 LIST OF TABLES Table co d23,0,0 I. Variation of 2j + l as a Function :1--o of the Order of the Approximation Page 39 I. INTRODUCTION In this thesis the exact solution of the problem of the diffraction of plane electromagnetic waves by an infinite slit of finite width in a perfectly conducting screen is discussed. The wave is normally incident and plane polar- ized with the electric vector parallel to the edge of the slit. The problem was first solved exactly using elliptic cylinder coordinates by Morse and Rubenstein1 in 1938. The solution involves an infinite series of Mathieu func- tions. The analytical prOperties of these functions are even now insufficiently understood and in the exploitation of the solution, one is led almost exclusively to numerical work. In our solution, we choose the circular cylindrical coordi- nate system to describe the electromagnetic field. In this coordinate system, the solution to the wave equation is an infinite series of Bessel functions of integer order multiplied by circular functions. We choose to represent the electromagnetic field by a contour integral in the complex order plane such that the infinite series of Bessel functions becomes the residue series of the contour integral. 2 In the circular cylindrical coordinate system, the boundary value problem is of the so-called two part variety and the boundary conditions lead to a dual set of homogeneous inte- gral equations that we attempt to solve by the Wiener-Hopf technique. This technique leads to an infinite set of equations in an infinite number of unknowns which we solve by successive approximation. The solution thus obtained enables us to verify all of the terms except one and, in addition, to obtain a new term in the expression for the transmission coefficient. This quantity was first obtained by a different approximation method by Sommerfeld2 (see, also, the review article by Bouwkampa). It is defined as the ratio of the power transmitted per unit area to the power incident per unit area. II. STATEMENT OF THE PROBLEM A plane electromagnetic wave, polarized with its electric vector parallel to the z-axis, is incident normally on an infinite slit of finite width, 2a, in a perfectly conducting screen. The slit lies in the xz plane with its length parallel to the z-axis; the screen extends to infinity both in the x and the z dimension (see Figure l). The time dependence is arbitrarily chosen as eiwt, there- fore, a plane wave of unit amplitude traveling in a given direction can then be written as e-i(k.r), where k is the 5 prOpagation vector. The quantity Ez shall, in this thesis, be denoted by the symbol U. The total electric field Ua above the xz plane will be written as U = U + U 3 y 0 (1) "V where U0 is the sum of an incident wave and a reflected wave as if there were no slit present in the perfectly conducting screen. The function U0 is written as follows: no = .11" - .‘m (2) UI is the perturbation in the total electric field caused by the presence of the slit. Below the xz plane, the total electric field Ub is just that caused by the presence of the slit, viz., Ub=UII; ygo (3) Hereafter, UI and UII will be called the scattered fields above and below the slit, respectively. A. Boundary Conditions The functions U and U I II satisfy the scalar wave equation, ViZee szr + kzy = o , (h) and are subject to the following conditions: (a) UI,II(x’y’2) a UI,II(-x’y’Z) ; (UI and UII are symmetric about the yz plane). (b) UI(x,y,z) = UII(x,-y,z) ; (symmetry about the xz plane). (C) UI=UII=0,¢=K/2,Pgao (d) a UII a U1 (e) UI=UII,p§a, 49:1/2. (r) a III II Lim ( ——-'—- + ikU P->°° 3? radiation condition). — U I,II) > O ; (Sommerfeld s (5) UI’ UII and their first derivatives must be square integrable over all points (x,y), (Meixner's edge condition). B. Discussion of the Boundary Conditions Boundary conditions (a) and (b) are statements describing the symmetries of the wave field which are the result of the geometry of the diffracting obstacle. Condition (0) says the surface currents in the conducting screen have such a direction as to cancel, exactly, the tangential component of the electric field incident on the screen. Continuity of the magnetic field in the aperture is contained in condition (d). Also, continuity of the scattered field in the aperture is contained in (e). Boundary condition (f) ensures that at great distances from 6 the slit, the scattered field represents a divergent travelling wave. The Meixner edge condition stated that only a finite amount of energy may be radiated by the singularity in the field at the edge of the screen (per unit length in the z direction). C. Integral Representation of the Scattered Field We would like to represent the solutions of the wave equa- tion by integrals of the form [LIA/\(M) H#(kp) on where HPkaU is a Hankel function and L is a contour in the complex order plane. However, in earlier work,h it was found convenient to discuss solutions of problems of this type for pure negative imaginary k, viz., k = -i1 , y > O, (5) because this puts milder restrictions on the choice of a contour and allows the use of the Lebedev transform theorem. After obtaining the solution of the problem for pure negative imaginary k, we transform to real positive k. In the light of the above discussion, we represent the scattered fields above and below the slit as follows: UI =fii°°MA(LL) cosMCP KAY?) dLL 3 4’ (6) Man HA HA mm U11: =fi wflAQL) cos{,u.(n - WINK fl(YP) d/(L, §§¢\(,u)/sin ye 8 is an even function of ,u. and analytic in a strip of finite width containing the imaginary axis. III. APPLICATION OF THE BOUNDARY CONDITIONS AND A DISCUSSION OF THE RESULTING INTEGRAL EQUATIONS In this section, boundary conditions (c) and (d) will be imposed on (6) and (7). These will lead to a pair of integral equations that will contain the unknown function Aw . It will also be shown in this section that it is possible to discuss the overall prOperties of [\Cu) on the complex Lt-plane such as the location of its singularities and its growth as I/LI —> on. This is possible without actual knowledge of this function by discussions involving the boundary values of the scattered field on the screen and in the aperture. Repeated use of Lebedev's transform theorem will be made in this discussion. Application of boundary condition (c) to (6) and (7) yields a homogeneous integral equation, viz., ice 490/0. A“) Igufif’) cos/55!- (1,“. = O ; P: a. (10) Application of boundary condition (d) to (6) and (7) yields 10 an inhomogeneous integral equation, viz., ioo f_i°°u2/\.(to sin 5-21 1am?) an e -yp; p g a . (11) The two integral equations, (10) and (11), are fundamental in the solution of the problem and represent the known boundary values on the screen and in the aperture. The complete representation of the boundary values is as follows: twat/Mu) cos ”‘1‘ Ku(YP) dJ-«L = U < a = 3" P = (12) O {9 Z a 100 _i°°u2.[\(p) sin’Eg-t- K“(YP) dpt = “'71" 3 F é a (1 ) _ 3{ Nu I 3 2 y f}: p where Uap is the unknown value of the scattered field in "V n! the aperture, and _ aUII { CyUII I } 374-33— is the discontinuity of the magnetic field across the xz ADI y—>O- - 2Y3" y->O+ plane (unknown for p 2 a). 11 A. Properties of JQJu) It is now possible to study the properties of [\XLO without actually knowing'j\XLO. In order to perform this study, we apply the Lebedev theorem to equations (12) and (15) giving the unknown function [\flu) in terms of the boundary values. By applying the Lebedev theorem to (13), we find, formally, that In: ’ Zya cos - ds (ix) AHA) = - 2 2 {K»(x) deJA- - n LL dK“(x) - 80,uflIX) dx x=ya - . IAN a 1°°s7f{3——4—UHI}K< )d (11+) RZLL a y u.Yp P ° Here, K“(x) is the McDonald function of complex order and (ix) is a Lommel function which arises in evaluating 504: a factor of the inverse Lebedev transform integral for the function «pspga O ; P': a ya I; KJx)dx. It remains to be proven that the conditions of the Lebedev viz., theorem are satisfied. Two of these, viz., g(O) = O, and the convergence of the first transform integral obviously 12 are satisfied by inspection of (13). The other conditions are that the inverse integral, viz., (lt), converge uniformly and that .5352.. ‘A&¥A&&l- is even and analytic in ill/LL = sin EA' 2 coslfié an infinite strip of finite width around the imaginary ,ic axis. That these are, indeed, satisfied will be shown later. Now it can be shown that the function inside the curly brackets of (l#) is an entire function of‘ja. The integral in the second member on the right side of (l#) is also an entire function of J4. First of all, IKAWC’HN rv e-Ye (0-1/2 as C’—-> 0° for all finite fl. Secondly, ”II I 1;;—-*- each satisfy Sommerfeld's radiation condition by virtue of boundary condition (f) and, therefore, also behave like {weed/2 as C —> m. Moreover, the integrand is continuous in 6 for all finite ,4, except at 6’ = a, the edge. However, bU/b y which here denotes the magnetic vector behaves as (9 - a).l/2 by virtue of the edge condition (3). Since Hflflve) is continuous at Q = a, the integral, although an imprOper integral, converges uniformly for all finite ‘;L-- it is entire inl/t. Note that the proof in the last paragraph also verifies one of the two validity requirements (viz., convergence of the inversion integral, not shown before) in the use of Lebedev's theorem above. Thus, (lA) gives us two properties of 1X9“): 13 (i) odd function of M. (ii) simple pole at LL = 0. We now consider (12). As it appears, the right side does not satisfy one of the conditions of the Lebedev theorem, viz., g(O) = 0, because the currents giving rise to the diffracted wave field must satisfy symmetry condition (a) and will interfere constructively along the z-axis, even in the aperture, 103., Uap(0) )5 O. The function UaP(YFU possesses a Taylor expansion about the point P = O which converges for 0 g P g a. This expansion is even in F) because of boundary condition (a), i.e., the scattered field is symmetric with respect to the yz plane and in the aperture, x is [9. Thus, we can write Uap (“y/J) ngga (15) Uap(0) + bapz + bit/)4 4» ... : where Uap(0) is the value of the scattered field at P = o. In order to circumvent the difficulty appearing in (12), we subtract the value of the scattered field at the origin from both sides, viz., firms/Mu) mtg-‘- K‘jyp) du- UaP = Uap(YP) - Uap(O) ; pg a (16) -Uap(0) ; p Z a 1% Now it can be shown that l .-. filiim cosL-‘g- KAY?) dpue (17) Using (l7), (16) can be written f” Hg (0) m -i°° LA{A(M) - fi } cos -2- KM(Y’0) M». a = V('Yp) (18) where Uapwp) - Uap(0) ; Pg a mp) = (19) -Uap(0) ; p _>__ a NY?) now has the proper behavior near P = O and one may use the Lebedev theorem to show that, formally, Ua (0) 2i sin‘-’:=2'5-fu up A0.) = 15;— + -—-——2——- o V(yl0)KH(yp) 7, . (20) It The validity conditions of Lebedev's theorem again require verification in this application. Considering (18), it is obvious that the condition g(o) = O is satisfied and also that the transform integral, viz., (l8), converges. We still have to prove that U (o) - .32... M = Aw) ““1 (21) Bin” 2 sin 521‘- is even and analytic in the strip and that the integral in (20) converges. This will be shown later. 15 The integrand in-(ZO) is a continuous function of f) in the entire infinite interval as follows from (19) and (15). As P ——> oo, VHF) remains finite and le‘YP) I ~e-‘YFP-l/2 for all finite Lt. However, near {0 = O, V(Yp)r~fiP2 and IKM(-{p)l~p"R°“’, therefore, the integrand in (20) converges uniformly in AA, only in the strip -2 < R8LL< 2 . At this point, notice that the result just obtained verifies the convergence of the inversion integral in the use of Lebedev's theorem on (18). The function of AA. represented by the integral in (20) can be continued outside this strip by a method whose steps we now outline: substitute the infinite series (15), and invert the order of integration and summation. This can be justified. Two types of integrals will result: 1: fim+l KM(YP) dF 3 m = 09 1e 29 co. , and -1 if) KM?) df) - Each of these integrals lead to wronskians of Ku(yp) and 82m+1,pL1YP)’ m a O, l, ..., and 80,p5ti°’ respectively. These expressions are analogous to the one in (14). Now, these Lommel functions all possess simple poles at }x.= I 2n, n = 1, 2, 5, ....7 The residues are unknown 16 because the coefficients in (15) are unknown, however, we do know where the singularities are located. Now the factor sin (532“- of the integral in (20) has simple zeros at Lt = 1 2n, n = l, 2, 3, ..., therefore, the product is an entire function of M. . Thus, we again see from (20) that (i) AHA) has a simple pole at p, = 0 (ii) A04) is an odd function of M. B. Growth of AHA) In discussing the growth of A01), we consider (14) for: (a) lRepJ —->°° and, (b) lImuI —-> on. (a) IReuI ——> on IAWI dso' (iya) 1 ' 27a cos (i;- "A 12,4. dK “(YBJ l —— + so,#(i'ya) d'ya + licos— ,ux —-—g- I I]: { #‘3 }Ke d? g ‘2“ KP(‘Ya) dso¥u(iya) M ' H- d‘ya l + "A I so diva) dKP(Ya) l [A d‘ya *B'%"fx()d.' (22) auYPP 9 where A and B M being the Now when ”L therefore, A180, I By the same I f: w ae |~ I it; Combining these results, we can write Thus, as K (va) K (Ya) l1l9u)| 5 A{ | ’“'3 + ’“ 2 ‘ .x& .1‘ + B l 53:8.) l . ,u. leuJ'-->‘w K’uflya) IAgxm/vl #2 l i cos/%; M i n2 , maximum absolute value of { BUII I } -S§——1- . is large7 3/2 o (iva)|~| —‘-E)——3| . ’1’“ 9/1. vat ‘1“ 9A ”“2 dK (ya) /u V I N l/C §m(ya)| e token, (23) (2h) (25) (26) (27) (28) (29) (30) 18 (b) IIm/al --> 0° Again taking absolute values of both sides of (14), we write _ L‘x . 2ya cos i? dso (iya) K 1 - ”VF” E I I2“ ' I (Ya) dya dKPfl‘Ya) 1 cos I? " sown”) 373—— + T! c)U . a{ -——I—Iil }K“(YP) dPl . (31) Now for large FTI, it can be shown9 I ( )I e-(x/2)rT| ( ) K ya ... 32 “ 171172 where 7' = Imp. Also, it is easy to see that ICOB —l~ e(1/2) '7' (53) Using (25) through (28), (32) and (33) in (31), we can write IAI~ A' + B' + C' (at) 171772 m572 m572 where A', B', and C' are constants (independent of’T). Thus, as I'TI —> ”9 IA |~ m'5/2 <35) Summarizing the properties of 1X1u9, we have (1) odd function of LL Ua (0) (ii) simple pole at LA ='O with residue Ii 19 ‘ K “(7a) (iii) lRepJ —> .e, lenni. ‘M __ I (1V) IIm/L‘l —> “’9 IAW) l~ rTI-S/ae T 3: ImI-L- C. Validity of the Applications of Lebedev's Theorem In section A, the proofs of the applicability of Lebedev's theorem to (12) and (13) were given except one, i.e., >\(u)/sinpx be even and analytic in the infinite strip of finite width containing the imaginary axis. This proof follows. Referring to the theorem and to (13), Nu) , Ml gm sinus 2cos‘2? Two of the preperties of AHA) already determined were that JAKLO is odd and has a simple pole at LL = 0. Therefore, F&nflk0 is analytic at LL.= O and is even in LL. The circular function in the denominator coséég is even in AL and has simple zeros at La = 1 (2n + l), n = O, l, 2, .... Since IAI~ITI'5/2 as m —> ..., |mu__l_ l~ -(x/2) ITI 2 cos (:2- |T|572 as [T1 -->‘~. Thus, it has been determined that Afil;fl%&; is even and 2cos E- analytic in a strip -1 < ReLk< 1. 20 Referring now to (18), U (0) U (0) >\(!2 = .[KPQ -' ‘22-- = [AJGNUJ - -2£I- Anni 51”,” 2 sin ’31 2 Msin ‘52-‘5- By arguments, in terms of the prOperties of A“), analogous to those above, it is easy to see that the numerator and denominator are both even in.pt, therefore, the ratio is even. Near L4 = O, U (o) {uA CO. Therefore, A1 - Ua (O) Lani 2 sin‘é} is even and analytic in an infinite strip -2 < Rel1< 2 containing the imaginary axis. IV. SOLUTION OF THE PROBLEM Before applying the Wiener-Hapf technique in the solution of this problem, it is necessary that we obtain two homogeneous integral equations over a contour L. The required prOperties of the integrands of these integral equations are: (a) that each integrand be analytic on a half plane and that these two half planes be complementary with a common strip of overlap containing the contour L, (b) that each integrand approach zero at least algebraically in all directions as the variable is allowed to go infinite in its respective half plane of analyticity. A. The Dual Integral Equations and the Wiener-Hapf Technique The two integral equations that we are concerned with are: finA(U) “03 8; KMYP) (1“ = 0 3 [9: a9 (10) and a impulsin‘fig-ACA) - 07w} KPWP) dpt = 0 ; f) g a , (36) where 601) is the Lebedev transform of the function 21 22 we: Ma 0 3 f): a ’ ViZo, . _g[129_3 - 12 {K (x) dso,L§ix) _ sinyx x2 “ dx dxufix) - Baum) 37—}... ° It can be shown by methods similar to those used in section III-B that CT(L0 is even in 11. and analytic in a strip sinux ‘ of width --1 < Reu< 1. As IRepI —> an, l 0.04) N sinpx KM(Ya) and as IImHl —> ”9 I 6- 0*) ~ e-(I/Z) '7' ”1-3/2 ,\, -ZZ__- sinpa where j” = Imlt. Now the integrand in (36) is an entire function of}4.. ‘We would like to modify it so that an infinite semicircle may be added to the contour (here taken along the imaginary axis but it can be moved since the integrand is entire) without changing the value of the integral. To accomplish this, we substitute, for Kyflvffl, the following identity, viz., I_ (7(3) - 1,1(YP) -; sinpfi With (37) substituted in (36), (36) becomes Iii-”HEE- {Main‘é} AU!) - O-(LAU IMdpL= 0; f): a . (38) 23 One can see that the integrand in (38) is still entire. Its decay on a right half plane is such as to allow the addition of an infinite semicircle without changing the value of the integral. In showing this, one needs the following relation, viz., u - I (1,0): F'g% {l + o(,-1;)} (39) as Re Lt—> +00. The remaining factor in the integrand has the following behavior, viz., eéL-{sein’gl Ado- 6 an. The product, therefore, behaves as 1(a)M but since < a, this product goes to zero. 1;. P- ( ) e(x/2)|’T| I y ... " P J at" IT": and L). W {Main-é- IAKW- 0(M)}~ -(u/2)VT| N ° sin(F(‘T)) 21: IT! where ’T’ = Imu, and - F(’T) is a real function of ’T . It is easy to see that the product behaves as sin(F(T)) : this T goes to zero for large I71. 24 ‘ Now that (38) defines a right half plane, we must be able to define (10) prOperly on a left half plane. In the present form of the integrand, this cannot be done. It is necessary to make the following "split": ,aAga) see/9:23 ,.. 6w. 9(7).) (to) where 994), obtained from (20), is the following expression: 69%) gfj: (vapor?) - Uap(0)} 17579) 96-9 . iU (0)}; ds_ (~iya) _EL—{%a.}{%.¥a) (1 13M _ 1: ya' '4' dI_ (Ya) s_1,/u_(-iya) _L-dya } (1+1) Analysis of (41) by methods similar to those used previously shows that 954%) is analytic to the left of the line Ros/4: +2, and that for large /a, it behaves as IIy‘éyaH. Utilizing these prOperties of 6}“), (10) can be modified and written as follows: Ana/(a) 94W?) got, = O; 6’ Z a . (#2) Now as Re/u—> duo, 65¢)N ( a 2)?“ and 9L(Y€)N fiat-7L"? (%fl2)flb , therefore, the product behaves as 11%: (CD/a7“). l +/% Since Re/u,< 0, this product goes to zero for G _>_ a as Re/L —> - me 25 ‘ 9 e+(1t/2)I”rl As llmul = ITI -—-> on, (M)~ ”11/2 and KM(yp)~ -(n/2)IT| ...e 172 sin(F(T)), therefore, the product again goes ITI to zero. Thus, we may add an infinite semicircle to the left of the contour and, since the integrand is analytic to the left of ReLA= +2, the value of the integral is not changed. Summarizing, we now have a dual set of integral equations satisfying the preperties stated at the beginning of this chapter and, also, a so-called functional relation connecting the two integrands. They are: f .4:— {nsnt-g: Am - C(11)}IJYP) dM= o : -i~ SinLL! pg 8. e (58) £1006 (M) KM(YP) dyL = O ; {J in, (1+2) and MAW) eoe‘tg- a 6040 +. 6(4)) . (#0) We now apply the Wiener-Hopf technique beginning with (38) by writing the integrand in terms of a ”plus function," i.e., a function that is analytic on a right half plane with algebraic decay in all directions on that half plane, viz., ‘* Assin‘tzuj\flu9 - (7(L9 PEI-2 2) { 2 sinut } 3 8+0» 3 (#3) 3*(UQ has a decay built into it which is at least of order 26 3L . Using (#0), (43) can be written 2 LL pa: + (a 2)“ tan -2' (em) +961») -6(LA) g (M) =fifiéfiT£ sinpcn } .(lH-t) We now define h-(VJ by . up. - 9(a) = _LLT&_7_( 8‘ f3, 1 -1L0u) (“5) or M jut One can easily see that h-OU) is analytic to the left of the imaginary axis with algebraic decay, e.g., a "minus function." Also, h-(—LD (a corresponding "plus function") would be analytic to the right of the imaginary axis and its value would be h-(w) ___ (Ya/2)"*I:'£b+n)_e_<-u> . (,7) when (45) and (47) are inserted in (nu), we get 1:15 - W = i W -r~n_ _ were (u) . 8 (,8, rUCUIsinps One more step is to write C7(L0 in terms of a "plus function" and a "minus function." We begin by writing 60¢) = Q1019 + Q20A) (1+9) where 27 (iva) a 80 pt Q1013 = - 3;; {I_“_(Ya) :Ya’ - dI_ (ya) - so,#£iya) dya (50) and ( ) iya a ds0,};- Q2(H) = 32¢; {I#('Ya) m - defiya) - 50,“.(iva) E??— . (51) Now, it can be shown that Q1 and Q2 are entire functions and that Q2(-p) = -Q1(p0. we introduce "plus" and "minus" functions into 0 (u) by writing -L& A calm) = 3% qi’ou) (52) QZuHJaZ) ago». (53) Notice also that q;(-u) = -qI(uJ. Now by inserting (49), (52), and (53) into (#8), there results i ii iii gen tan _ 2L!- - + 2 - (YaZZ) h (~I-A) s (n) = { )h (u) - n 21" (IQFU +M) cos2 LEE iv v qim) (Ya/2)“ qgm) M n - . (51+) ['20. + l)sinp.x Considering (5#), we would now like to collect plus terms on one side and minus terms on the other. Before doing this, 28 we must discuss where the poles lie in each of the terms (i), (ii), (iii), (iv) and (v) in (SA): (i) is an entire function of LL. having algebraic decay in all directions, (ii) -- remove its simple poles at LL 2 -1, -3, ..., -(2n + l), n = O, l, 2, ... by subtracting an infinite partial fraction series; then this function minus this partial fraction series will be a minus function to the left of ReLA= +1, (iii) -- remove simple and double poles at Lu'z (2n + l), n = O, l, 2, ... by infinite partial fraction series, then this function minus its partial fraction series will be a plus function to the right of Rep: -1, A (iv) is a minus function to the left of _Retms o, (v) -- remove simple poles at LL = O, l, 2, ..., then this function minus its partial fraction series will be an entire function of;x.. When each term in (SA) is modified according to the preceeding prescription, "plus functions" are collected on one side of the equation and "minus functions" on the other. The properties of the "minus" side of the resulting equation are (l) analytic to the left of the line ReLL= O, (2) algebraic decay in all directions in that half plane. The prOperties of the "plus" function side are 29 (1) analytic to the right of the line ReLx: -1, (2) algebraic decay in all directions in that half plane. Thus, the "plus" side and the "minus" side have a common strip of overlap, each side is analytic in its respective half plane and has algebraic decay in that half plane, therefore, by Liouville's theorem, each side is equal to a constant, zero. Letting LL.= -2m - l, m = O, l, 2, ..., the minus side of the above discussed equation (drapping the superscript) can be written 2 l h(-2n - l) -—h'(-2m-l)+-—Z 1‘2 1‘2 n=0 n - m min 1 (ya/2)hn+2h(-2n - 1) “"72 2 + 21 n=O (2n):(2n + 1)£(m + n + l) hn+2 _g- 2 (YRZZ) h(-2n - 1) y {108(E58') - + :2 n=O (2n):(2n + l):(m + n + l hn+2 l ]. (vs/2) h'(-2n-l) -(p(2n + l) ' En + 2} - IZDBO (2n)1(2n+l)2(m+n+l) g _ q(2m . 1) _ (~1)nq(n)(va/2)2n (55) ’ u(2m + 1) 2 n=0 x(n!) (2m + n + 1) As can be seen by inspection, this infinite system involves the unknown h(-2n - l) and its derivative. It will be seen in the next section that the scattered fields can be 3O written in terms of this unknown. B. The Scattered Field Expressed as a Series Expansion Referring to (7), the scattered field, U for y g 0 II’ (below the slit) is m UH a $0qu cos[M(1t - |¢|)}K“(YP) an . (7) We wish to express this integral representation as a residue series. To accomplish this, it is necessary to use (#0), 712. 9 MAW) eos‘éé‘l = em) + 9am . (1+0) Inserting (40) in (7) yields f“ 90.) cos{p\(u - |¢|).}K“(yp) on ”I: = 2 -i°° ILL cos 2! . (56) Since 9(1).) is analytic to the left of Re u: +2 and behaves as |I_L£va)l, an infinite semicircle can only be added on a left half plane. The poles of the integrand are the simple poles of cos (i;- which occur at Ix = :(2p + l), p = O, l, 2, .... Therefore, applying the Cauchy Residue Theorem yields n on = -81 D; e(-2n - 1) K2n+1(vp)(-l) . . cos{(2n + l)|¢l} . (57) Now 31 (ika 2)2n+1h(-2n - 1) 6(-2n - l) = - —Lr(2n + l) (58) and K(ik€’) e - 951‘- e‘(i“/2)(2n*i)n Héillme) . (59) Substituting (58) and (59) into (57), UII becomes Z (-1) n(ka/2)2n+1h(-2n - 1) = I” r‘(2n + 1) n=0 . cos{(2n + l)|¢|]H(2)l (he) . (60) Note that the transformation back to real positive k has been made via Y = ik. C. The Transmission Coefficient The transmission coefficient,‘T, is defined as the ratio of the power transmitted by the slit per unit length along 2 to the incident power on the slit per unit length along 2. In terms of the complex Poynting vector, it can be shown that this quantity may be written bu 7.... - ifs/2 Re(iUII 3-52 )€d¢ . (61) When equation (60) is substituted into (61) and the integral is evaluated for large p , the following expression is obtained: an 7': 2u2ha{|h(-l)l2 + Egfi- |h(-3)|2 + ...} . (62) Reference is now made to equation (#6), viz., 32 t M . h(M98) = C F(1 -i)e(‘*1€) (#6) where c = 2: , k = -iY' . Now it can be shown that equation (#1) can be expressed as a double series, viz., 2p 19 2j-M2 c 6(MQC) =3 2nj=o bZJc P30 (-#+2P+23)P:_T—M P+l" (63) where the sz are the same as in (15). Combining (#6) and (63), we arrive at ( ) i Z ( )2 ‘Zpd (6#) h M-c = - d c ,. 9 at 3:0 23 on (2p-u¢23)p£(l-P9p where _ 2j-1 d23(c) - szs and (a)p = «(a + l)(a + 2)...(a + p - l) .0 10:1, p=0’ 19 see. Setting AA = -2n - l in (##) yields h(-2n - 1,5) = '3‘- }: d2 (5) ' 2x 380 j Z r ‘Zpd (6 ) e . 5 P=0 2p + 2j + 2n + 1)p2(2n + 2)P 35 Taking the derivative of (6#) and letting Lk = -2m - 1 gives c2p+l 2%J ==JL-§Efl j(c) d'lpa-Zm-l 2n 3:0 p=0 p3(2p+2m+2j+l)(2m+2)p e{ 1 ...—L... 2m + 2j + 2p + 1 2m + 2 + + + 1 } (66) 2m + 3 "' 2m + l + p ° When (65) and (66) are substituted into (55) and the right side of (55) is expanded in terms of powers of c, there results the following expression: 1 22 2p+l ' :3j=0 d23(°) P=0 p1(2p + 2j + 2m + l)(2m + 2)p . . {2 1 + —1_ + ——— + . . . + p + 2j + 2m + 1 2m + 2 2m + 3 l } + 2m + l + p i Z 1 Zodz p20 c2p+l + 3 n - m j 2p+2n+l+2j)p. (2n+2) - 2K n:O =0 P m c#n+2 on --3*- 220 2 E: d. (c) ‘ M3 W: (2n)!(2n + l)1(m + n + 1) 1:0 2j 29+! "F\/18 p= =0 (2p + 2n + l + 2j)p:(2n + 2)p + c#n+2 .1. E:OT :1 {lo c - “3 W: fin) (2n + l)' (m + n + l) 8 5h (P(2n+l)-EE—;—}J:oda j(c). . c2p+1 i=0 (2p + 2n + 1 + 237§:(§n + 27; 4n+2 ' ‘1‘: (27:?2 m c 1 )2d (c) ‘ 213 n=0 n n + m + n + 1 jgo 23 Z c2p+l . (2p + 2n + 21 + 17(2n + 2)p p= =0 p' .{ 1-+_-_1-—+ .00 + 1 )3 2p + 2n + 23 + 1 2n + 2 2n + l + p is m 1 (2m + 1)1:2 {(1 - 61390) E IZn + 15 - 1} " 3 16 2m-1 1 2(2m + 1);2 {- 3(2m + 2F "5 (1 - 5 O) . 1 (1-31)(2n-2) ' {m(m + l) (m + 2)(m1+ l)m(m€ '17} + "°} ° (67) We now seek a solution of (67) for the d2j(c) valid at and in the neighborhood of c = 0 by successive approximation. Comparison of the magnitude of terms in (67) leads us to conclude that d2 (c) must have the following form: 3 d (c) = d + :2{(log c - q:(1) - 1)d + Zj 2j,0,0 2 Zj ,1, O h l 2 d23,1,1} + 8 {(log 5 — g)(1) - 2) d2j,2,0 + + (log a 4(1) - lm ' ' + 35 2 23,2,1 d23,2,2} * €6{ coo } + co. 9 (68) Before proceeding further in the solution of (67), let us utilize (68) to obtain a more appropriate expression for the transmission coefficient. First, the transition back to real k is made via y = ik and since c = Ya/Z, this is equivalent to replacing a by ika/2. With this modifica- tion, one can write (6h), using (68), as follows: hsuqka) a2 d23,0,0 + 330 (23 + 2 -,a)(1 ng) a . +(longa-é-4-i‘); —J-'—'-2 10 + ... + 3 ... =0 (2 -,u. + 2j)(l «703 + _§1;§;§ } + 0(k7a7) (69) where q)(l) = - log 8. When (69) is squared and pk takes on the values -1, -3, -5, ..., the transmission coefficient can be written as follows: .(Zgam,(zm,+1(zm). 3:0 3 + l 3=0 23 + 1 2 3:0 23 + 3 .(2 1211252)“: fawn}: Sam)“ 3:0 23 + 1 3:0 23 + 1 3:0 23 * 1 7 7 Ail-{(10%- EEK-)2“: 333131214. 42% )1(§od 23,1,0 3:02 23 + l 1 l 2 0 2 1 0 + (log%—- 2)(2(3:0 Eli-T)(Zod 5311—?)4- 57 + “NIH + (:0 d23,o,o)(§Od 23,2,1) 23 + 1 23 + 1 d d 21.1110 :0 211131 (HZOZJ+1)( 23+1)}+ooo}+ + + In order to obtain expressions allowing solutions for the unknown d's, we substitute (68) into (67) and set coef- ficients of like orders of magnitude in c equal to each other. Only the systems of equations giving solutions for the d23,o,o’ d23,l,0’ d2j,l,l’ d23,2,0’ and d23,2, 1 "ill be listed: .. .. Eda. { 1 “1-2 1 }= i=0 3’0’0 (23 + 2m + 1)2 2n=0 (n-m)(23+2n+1) aim x m x - - (§;_:_T) {(1 - sm,0) 2m(m + l) - 1} - E’Sm,0 (71) Z. { 1 J): 1 }= 3:0 23.1.0 (23 + 2m + 1)2 2n_ -0 (n-m)(2n+23+1) nfim d 1 2 0 O =m+l Zj+1 (72) 38 co D\/18 1 l2 1 }_ - 2 (n-m)(2n+23+l) - d23,l,l{ i=0 (23 + 2m + 1)2 n=0 nfim - E: d23,o,o { 1 (21 2 2 g ” 3:0 (23 + 2m + 3)‘E3_:—2 08 - - 3 - °" 2 1 1 1 1 -2j+1)-Fm(l+-2-+-3-+...+E)}- co _ n , _ 1 d21,0,0 _ n + 8(m + 1)2 2(m + 1) 3:0 (23 + 1)2 8. m,0 + I g + I {_ (2m - 1)- EH m,l 2(2m + l) 3(2m + 2) - 2 m(m + l) - m,l (n+2)(m+l)m(m-1) (73) 2 a { 1 A): 1 }= 3:0 23’2’0 (23 + 2m + 1)2 2n=0 (n-m)(2n+23+l) nfim —- 1' 2: 3211119 (7#) " m + l 5=0 23 + l 59 d Variation of Z iii-212 23 + l 3:0 as a Function of the Order of the Approximation 4x# 8x8 d0,0,0 = 2.0187102 40,0,0 = 2.005408# 62.0.0 = -1.0672165 42,0‘0 = -0.89439#63 9 ? - - d4 0 0 = 0.202h5745 dh,o,o = -2.5352520 ? ? l . d6,0,0 = -°-735635#9 d6,6,6 = 11-119179 d89090 8 .#3'9h5776 d d . 66.2669 4 5311-919 = 1.5983721 1°'°'.° 3 i=0 d12’0’0 a -67.7183h5 d14’0,0 = 12.717386 6x6 7 . . _ d d090g0 - 2.0077678 320 #2: +010 I 1.530171 d2,0,o . -1.058509o d4’0,0 = 0.35200261 d6.0.o = -2.7605063 d8,0,Q 3 “01"203099 d10,o,o = -2.7091091 5 d 23,0,0 3 21 + 1 1.5825320 i=0 #0 10x10 do;o;o = 2.002967# d2,0,0 = -1.1958266 dines) = 3.8505777 de’o’o = -28.299519 d8,0,0 = 81.291771 d10’0,o = -87.507227 d12,0;0 . -38.883055 d1h.0.0 s 163.80%38 d16,0,0 = -12#.17881 d18,0’0 = 29.265336 9 £28 ggligfg 8 1.57385 12x12 do;o;0 = 2.0026802 d2’0,0 = -1.0015692 d#.0.0 a -0.303958h3 66’0’0 = 1.1861691 d8,o’o = -7.7028185 d10’0,o = 18.972307 d124040 a -2h.769373 d1h4040 = 25.0h0875 dlé’o’o = -26.08380h d18;040 = 7.563h3u1 d20,0,0 . 17.h76718 d22,o,o . ~12.216622 11 , 2 12.1.2.9:2 = 1.57#942 3:0 23 + 1 60,040 - 2.0025757 d24040 = -o.987838#2 d#,0,o = -0.53010458 d64040 = 2.1633397 d8’0,0 = -7.5h75131 010,0,0 = 9.1255387 d12,o’o = -2.0882360 d1#,0,0 = 6.0645652 d16,0,0 = -16.#36335 d18,0,0 s —12.h38990 620,0,0 = u3.091#01 622;0;o = -20.321572 dah’o’o = -3.8012758 626,0;0 = 1.8655891 1 2311919 . 1.57h912 3:0 3 + 1 16x16 d0,0,0 d2,0,0 d9,0,0 d6,0,0 d8,0,0 d10,0,0 d12,0,0 d11+,0,o d16,0,0 d18,0,0 d20,0,0 d22,0,0 d29,0,0 d26,0,0 d28,0,0 d30,0,0 15 d i=0 2.0023806 -0.97961759 -o.65507263 2.5709285 -6.9332251 5.65507ot -2.6968822 15.213107 -2#.266l56 36.726h81 -31.9hh528 31.938553 18.111970 -52.795791 21.888138 £112.19. = 1.5711668 #2 E: d2' 2 1 { 1 2 ' %’ I )(21 2 1)} 3 3:0 3’ ’ (23 + 2m + 1) n=O 3‘” 3+ 3* nil _ _ 1 Z d23,1,0 + 1 Z d23,1,1 _ ‘ 23 + l m + l 23 + l #(m + l)2 3:0 3:0 _ 1 d23,1,0 . (75) at“ 1 15 3:0 (23 + 1)2 In all of the above expressions, m = 0, +1, +2, +3, .... It is to be pointed out that the solutions for d , 23,0,0 and d are identical. d23,1,0 23.2.0 One also notices that the matrix of the coefficients of all the d's is the same for each expression, i.e., (71) through (75). The method used to solve (71) and (73) was that of successive approximations. The order of the matrix and its corresponding solution is listed in Table I. d _ From this table, one notices that the E: 2%11212 appears 380 to approach x/2. This value is necessary in order to have agreement with Sommerfeld's resultz. (See also Bouwkampa). as 12.1.2.2 Granting that 23 + 1 approaches x/2, then also do 3:0 43 co ‘ d d . ’ 2 110 2 2 Z 53-11-4— amid:0 gilt—1’:- approach n/2. i=0 In obtaining a solution to (73), it was necessary to use the numbers d0,0,0’ d2,0,0, dh,0,0’ .... But as can be seen from Table I, the only numbers that were determined were d0,0,0 and d2,0,0 perhaps to the tenth place. Thus it was not possible to obtain even an approximate solution to the first coefficient d . . 0,1,1 However, using the results that have been obtained above, one can express the transmission coefficient, as far as we have gone in the expansion, as 233 255 T=1‘—‘g‘%- - 1%fi-<1°s%‘-‘--%> + 77 ka §_§i§_— (103 2%5 - %02 + unknown terms . V. DISCUSSION AND CONCLUSIONS The Lebedev transform, when applied to the problem of the slit, yields two homogeneous integral equations. These two equations are solved by the Wiener-Hapf technique resulting in a double infinity of linear algebraic equations in terms of the unknowns h(-2n - l) and h'(-2n - l), n = O, 1, 2, .... _The method of successive approximations was applied to these equations. An expression was obtained for the transmission coefficient involving the unknowns h(-2n - l) and h'(-2n - 1). It was possible to obtain the first few terms in this expression. There was exact agreement between our result and the known result5 except for a constant which we have not been able to ascertain because of the slow convergence of the series involved. However, a new term has been obtained. an 1. 2. 5. 7o 80 9. REFERENCES Morse, Po Mo, and mlbenstein, Po Jo, P1118. Raye, 1958, a, 895. Sommerfeld, A., Vorlesungen Uber Theoretische Physik, 1st ed., (Wiesbaden: Dieterich, 1950),‘fl. Bouwkamp, C. J., "Diffraction Theory," Reports On Progress In Physics, l95#, 12, 55. . Oberhettinger, F., Comm. Pure and Appl. Math., 1954, Z, 551. Lebedev, N. N., "Sur Une Formule D'Inversion," Comptes Rendus (Doklady) de l‘Academic des Sciences de 1'URSS, 1946, 22, M8. Erdelyi, A. et al., Higher Transcendental Functions (1955 McGraw-Hill), g, p. 90, (7). Ibid., p. #0, (69). Erdelyi, A. et al., Tables Of Integral Transforms (1954 McGraw-Hill), g, p. 175, (2). Leitner, A. and Wells, C. P., "On The Radiation By Disks And Conical Structures," Departments of Physics and Mathematics, Technical Report No. 1, Detroit Ordnance District Contract No. DA-20-018-ORD-1355h, August 1955, Michigan State University, East Lansing, Michigan. 45 MICHIGAN STnTE UNIv. LIBRQRIES \IHI"INN“\ll‘llllllmml“mm\IHWIWIHNHWIVI 31293017430087